Chapter 1 · Rational number foundations
Order of Operations with Integers and Finding Errors
Evaluate integer expressions one operation at a time in the agreed order, then check worked solutions line by line to find and correct mistakes.
A numerical expression is a string of numbers and operation signs, such as . To evaluate an expression means to find the number it equals. That number is called the value of the expression.
In this lesson the numbers are integers: the counting numbers , their opposites , and . You already know how to add, subtract, multiply and divide two integers. Now you will combine those skills in longer expressions. Then you will use the same thinking to find and fix mistakes in someone else's written work.
Why does the order of the steps matter? Look at . The part means two copies of . That is one quantity, , and the expression asks you to add it to . So the value is . If you added first, you would split up the two copies and get , a different number.
So that an expression means the same thing to everyone, there is an agreed order of operations. For an expression made of integers, grouping symbols and the four operations (addition, subtraction, multiplication and division):
- Work inside grouping symbols, such as parentheses, first. Inside a group, follow steps 2 and 3.
- Then multiply and divide, moving from left to right.
- Then add and subtract, moving from left to right.
Parentheses around one negative number, as in , only keep the sign with its number. There is no operation inside them to carry out.
Each step of an evaluation uses a fact you already know about integers. A few words first. Numbers that are multiplied are factors, and the result is their product. In a division, the dividend is the number being divided, the divisor is the number you divide by, and the result is the quotient. Nonzero means not equal to zero. The additive inverse of a number is its opposite, the number you add to it to get . For example, the additive inverse of is , and the additive inverse of is .
Each fact in the table is true only under the condition written with it, so read the two together.
| Situation | Fact | Example |
|---|---|---|
Subtracting | Subtracting a number is the same as adding its additive inverse. | |
Multiplying nonzero factors | An odd number of negative factors gives a negative product. An even number gives a positive product. | and |
Multiplying when a factor is zero | Zero times any number is zero. | |
Dividing a nonzero dividend by a nonzero divisor | Same signs give a positive quotient. Different signs give a negative quotient. | and |
Dividing zero by a nonzero divisor | Zero divided by a nonzero number is zero. | |
Dividing by zero | Division by zero is undefined, so the expression has no value. | is undefined |
Key concept
One operation at a time, in the agreed order
An expression made of integers, grouping symbols and the four operations has exactly one value, as long as no divisor is zero. To find it, replace one operation at a time with its result: grouping symbols first, then multiplication and division from left to right, then addition and subtraction from left to right. You are swapping an operation for a number equal to it, so every correct line has the same value as the line before it. An error is a line where the value changes.
Worked example
Multiplication before addition
Evaluate .
Decide the order. The parentheses hold only the number negative 3, so there is nothing to work out inside them. That leaves an addition and a multiplication, and multiplication comes before addition.
Multiply 6 by negative 3. Both factors are nonzero and exactly one of them is negative. One is an odd number, so the product is negative. Since 6 times 3 is 18, the product is negative 18. Replacing the multiplication with a number equal to it keeps the value of the whole expression.
Add negative 8 and negative 18. Both numbers are negative, so the result is negative, and it is 8 plus 18, or 26, units from zero.
Answer
The value of is .
Worked example
Grouping first, then left to right
Evaluate .
Start inside the grouping symbols. The parentheses at the end hold a subtraction. Subtracting 8 is the same as adding its additive inverse, negative 8, and 3 plus negative 8 is negative 5. Replace the group with negative 5.
Next, multiply and divide from left to right. The division is on the left, so it goes first. The division sign applies only to the 6 that follows it, so multiplying 6 by negative 2 first would change the value. The divisor 6 is not zero, and negative 24 and 6 have different signs, so the quotient is negative. Since 24 divided by 6 is 4, the quotient is negative 4.
Multiply negative 4 by negative 2. Both factors are nonzero and both are negative. Two is an even number of negative factors, so the product is positive. Since 4 times 2 is 8, the product is 8.
Only a subtraction is left. Subtracting negative 5 is the same as adding its additive inverse, which is 5.
Answer
The value of is .
Now turn the idea around. In correct work, every line has the same value as the line before it. So when a final answer is wrong, at least one line changed the value. Your job is to find the first line that did.
Check one line at a time. See what the writer changed from the line before, and ask two questions:
- Right operation? Is it the operation that the order of operations puts next?
- Right result? Is the number written for it correct, including its sign?
If both answers are yes and the rest of the line is copied unchanged, the line still has the same value, so move on. When you reach the first line whose value is different, say what went wrong. Most often it is an order error (an operation done before its turn) or a calculation error (the right operation with a wrong result, often a wrong sign). Correct that line, then redo every line after it, because the writer built those lines on a wrong number. The table shows this check on one student's work.
| Line | Student wrote | Check | Verdict |
|---|---|---|---|
Given | This is the expression to evaluate. | Start here | |
1 | Grouping goes first, and is correct. | Correct | |
2 | The student did first. But the is part of the product , which comes first. | First error (order): the value changes from to | |
3 | is correct arithmetic, but it uses the wrong line 2. | Must be redone | |
2, corrected | Multiply before subtracting: . | Correct | |
3, corrected | Subtract: . | Correct value |
Worked example
Find and fix a sign error
A student evaluated and wrote four lines. Line 1: . Line 2: . Line 3: . Line 4: . Find the first line with an error, say what went wrong, and correct the work.
Check line 1. Grouping symbols go first, so working out 3 minus 9 was the right choice. Subtracting 9 is the same as adding its additive inverse, negative 9, and 3 plus negative 9 is negative 6. Line 1 is correct.
Check line 2. Multiplication and division come next, and the multiplication is on the left, so the student chose the right operation. But the result is wrong. Negative 4 and negative 6 are nonzero factors and both are negative. Two is an even number of negative factors, so the product is positive 24, not negative 24. Line 2 holds the first error: a sign error.
Look at the later lines. Line 3 divides 18 by negative 6. The divisor is not zero and the two numbers have different signs, so the quotient negative 3 is correct. In line 4 the student subtracts negative 3 by adding its additive inverse, 3, which is also correct. These lines add no new error, but they use negative 24, so you must redo them.
Redo the work from line 2 with the correct product, 24. The division that follows gives negative 3, as the student found.
Finish with the subtraction. Subtracting negative 3 is the same as adding its additive inverse, 3.
Answer
The first error is in line 2, a sign error: the product of the nonzero negative factors and is , not . The correct value of the expression is .
Takeaway. An expression made of integers, grouping symbols and the four operations has one value when no divisor is zero. You find it one operation at a time: grouping symbols first, then multiplication and division from left to right, then addition and subtraction from left to right. For each operation, use the integer fact that fits and check its condition. Because every correct line equals the line before it, you can find a mistake by locating the first line where the value changes, naming what went wrong, and redoing the work from that line on.